Star-Delta and Delta-Star conversion concept is very basic but most important concept for Electrical Engineers. Be it day to day calculation or solving some numerical of competitive exam, we definitely need this conversion. In this post we will be discussing this Star to Delta and Delta to Star Conversion.

For understanding this conversion, we will split our discussion into two parts, Star to Delta Conversion and Delta to Star conversion.

Delta to Star Conversion:

For Star connected load, the total impedance between terminals A and B,

= Rb+Rc

But for Delta connected load,

Total impedance between terminals A and B is Rbc parallel to series combination of Rac and Rab.

So, Total impedance between A and B = Rbcx(Rac+Rab) / [Rbcx(Rac+Rab)]

Therefore for the equivalence of Star and Delta, the total impedance between the terminals A and B for both the system must be same.

Hence,

Rb+Rc = Rbcx(Rac+Rab) / (Rbc+Rac+Rab) …………………(1)

Similarly, for total impedance between terminal B and C we can write,

Ra+Rb = Rabx(Rbc+Rac) / (Rab+Rbc+Rac) ……………………(2)

For total impedance between the terminals A and C,

Ra+Rc = Racx(Rab+Rbc) / (Rab+Rbc+Rac) ……………………..(3)

Now for finding Ra, Rb and Rc, we add equation (1),(2) and (3),

2(Ra+Rb+Rc) = 2(RabxRbc + RbcxRac + RacxRab) / (Rab +Rbc+Rac)

So,

Ra+Rb+Rc = (RabxRbc + RbcxRac + RacxRab) / (Rab +Rbc+Rac) ………(4)

Subtracting equation (1) from (4),

Ra = RacxRab / (Rab +Rbc+Rac) ………………(5)

Subtracting equation (2) from (4),

Rc = RbcxRac / (Rab +Rbc+Rac) ………………(6)

Subtracting equation (3) from (4),

Rb = RabxRbc / (Rab +Rbc+Rac) ……………..(7)

From the above expression, it can be concluded that

Star R = Product of Delta R connected to the Same Terminal / Sum of R

Mind that the above transformation from Delta to Star has been brought out with taking Resistance while the same is true for Impedance Z.

Star to Delta Conversion:

As we can see from the figure above, equation (5), (6) and (7) are still valid for Star to Delta conversion but the only difference is that, this time we need to find Rab, Rbc and Rac.

So, we will rearrange equation (5), (6) and (7) as below.

Ra x (Rab+Rbc+Rac) = RacxRab ……………….(8)

Rb x (Rab+Rbc+Rac) = RabxRbc …………………(9)

Rc x (Rab+Rbc+Rac) = RbcxRac …………………(10)

Dividing equation (8) and (9),

Ra / Rb = Rac / Rbc

So, Rac = (Ra x Rbc) / Rbc

Similarly by equation (9) and (10),

Rb / Rc = Rab / Rac

So, Rab = (Rb x Rac) / Rc

Now, putting the value of Rac and Rab in equation (8), we get

RaRb + RbRc + RcRa = RaRab

So,

Rab = (RaRb + RbRc + RcRa) / Ra

Similarly,

Rbc = (RaRb + RbRc + RcRa) / Rc

and

Rac = (RaRb + RbRc + RcRa) / Rb

Mind that while converting Star to Delta, connect the impedance between, say at terminals A and B, so that impedance will be (RaRb + RbRc + RcRa) divided by remaining one impedance connected to the remaining one terminal. So, see when I am calculating Rab, the remaining one impedance is Ra so I divided (RaRb + RbRc + RcRa) by Ra.